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7 votes

Characters with all higher exterior powers irreducible

For an irreducible of degree $d$ to have this property, the sum of the squares of the $\binom{d}{i}$ with $2i\leqslant d$ has to be at most the group order. Now look at faithful irreducibles of $S_n$ ...
Dave Benson's user avatar
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2 votes

What is known about finite dimensional modules over the nilCoxeter algebra?

Motivated by the answers I tried to check when it is symmetric and I got the result that it symmetric is if and only if conjugation by the longest element acts as the identity in the corresponding ...
Mare's user avatar
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3 votes

What is known about finite dimensional modules over the nilCoxeter algebra?

@DaveBenson, has already given a beautiful answer to this question. I just wanted to point out that a number of the things he says (although not his full computation of the dimension of the Ext-...
Benjamin Steinberg's user avatar
10 votes
Accepted

What is known about finite dimensional modules over the nilCoxeter algebra?

This algebra has just one isomorphism class of simple module - let's call it $S$. Its projective cover is the regular representation, and is also the injective hull. The socle of the regular ...
Dave Benson's user avatar
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6 votes

Conjectures in the representation theory of the symmetric group

Here are some of my favorites, some were mentioned in comments already. I'm not going to be too picky about the distinction between conjectures vs open questions. The Saxl Conjecture was already ...
Nate's user avatar
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12 votes

Conjectures in the representation theory of the symmetric group

Just a few things that come to my mind; most are about the base ring $\mathbb{Z}$, but a good answer would tell us something interesting about $\mathbb{Q}$ as well. Specht modules can be defined not ...
darij grinberg's user avatar
5 votes

Conjectures in the representation theory of the symmetric group

There are several conjectures in this area that pertain to computational complexity. Most well known are those that arise from geometric complexity theory à la Mulmuley and Sohoni. The original ...
Timothy Chow's user avatar
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5 votes
Accepted

Asymptotic character theory of unitary groups via shifted Schur functions

Yes, they probably mean https://arxiv.org/abs/q-alg/9709011 There, they consider the Jack generalization of the problem, but if you set $\theta=1$, then you get the theory of characters of the unitary ...
Leonid Petrov's user avatar

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